How far away must a galaxy be for its light never to reach us due to the expansion of the universe? My understanding is that at the present rate of expansion of the universe some galaxies are growing more distant from us at such a rate that light from them will never reach us. My question is how far away a galaxy must be for this to be true of it.
 A: In this paper by Gott et al., on p. 466 they define the "future visibility limit", saying in the published version that "No matter how long we wait, we will not be able to see farther than this", and on p. 7 of the arxiv preprint they similarly say "If we wait until the infinite future we will eventually be able to see the Big Bang at the co-moving future visibility limit. Stars and galaxies that lie beyond this co-moving future visibility limit are forever hidden from our view." The text under Fig. 1 on p. 467 of the paper (p. 43 of the arxiv preprint) gives the future visibility limit as 4.5 times the Hubble radius, mentioned on p. 465 (p. 7 or the preprint) to be 4220 Mpc, which is 13.76 billion light years. So if the future visibility limit is 4.5 times that, it should be about 62 billion light years, which matches with where the cosmological "event horizon" cone hits the "comoving distance" axis in the third diagram in fig. 1 in this paper by Davis and Lineweaver (the text underneath says that "Our event horizon is our past light cone at the end of time, $t = \infty$ in this case", so evidently this is the same as the future visibility limit).
Note that this may actually not be exactly what you're asking, because the diagram in the second paper indicates that the event horizon they're talking about is the comoving distance such that we will never see light emitted from that location at any point in time, even arbitrarily close to the Big Bang. (If you're not familiar with the term, comoving distance is defined so that an average pair of galaxies will have a fixed comoving distance, basically leaving out the expansion of space, and it's also defined so that the comoving distance to a galaxy at present is identical to its present proper distance, i.e. the distance that would be measured by a series of small rulers laid end-to-end at the present cosmological time.) You may have been asking instead about the comoving distance such that if light was emitted from that location now, it would never reach us. I don't have a paper that gives a precise number for this distance, but looking at where the event horizon (the past light cone of our location at $t=\infty$) intersects the "now" axis in the third diagram in Fig. 1 of Davis/Lineweaver paper, along with the pretty much identical diagram created by Pulsar that appears in Christoph's answer, if you use a drawing program to draw a straight line between the intersection point and the "comoving distance" axis, it appears the answer is very close to 16 billion light years.
A: In addition to the answers already given:
You have to differentiate between the galaxies whose light which they emit today can be received by us in the future, and the galaxies whose light which they emitted in the past all the way back to the big bang that we can receive today and in the future.
The first one is the cosmic event horizon (16 Glyr), which is currently a little farther than the Hubble radius (in a few billion years when the matter density will become neglible compared to dark energy those will be the same). It converges to a proper distance of about 18 Glyr, while its comoving distance is shrinking.
That means that galaxies behind the Hubble radius (14 Glyr), but within the event horizon (16 Glyr) with a recessional velocity greater than c can still send us a signal that we will be able to receive in the future. In later times both radii will converge to 18 Glyr.
The second is the particle horizon (46 Glyr), that one has an exponentially growing proper distance, while its comoving distance converges to 64 Glyr.
That means the objects whose light that we receive today was emitted soon after the big bang are now 46 Glyr far away, while later we will also see the early childhood of objects which are currently 64 Glyr away.
The reason why the proper distance of the event horizon and the comoving distance of the particle horizon converge to a fixed value in the future is because the Hubble parameter is also converging to a constant value due to the accelerated expansion (cosmological constant), which will be about H0√Ωλ, which is about 83% of today's expansion rate (also see here).
That can be seen on the proper distance versus comoving distance diagram, for a comparison see here (it looks a lot like the image in the other answer, but the time axis is linear, while in the image shown in Christoph's anwer the curves are flattened out by using a conformal time axis).
A: If our ideas about cosmic evolution are correct, galaxies that are visible today will in principle remain visible in the future. As time goes on, light from more and more distant galaxies will be able to reach us, and the number of observable galaxies will increase.
However, there exists a cosmic event horizon$^1$, so this is an asymptotic growth: There's a maximum number of galaxies a future observer will in principle be able to detect.
The problem is that redshift will make receding galaxies appear fainter and fainter, so in practice, you will see fewer and fewer galaxies, and someday you will be unable to detect distant galaxies at all. This is due to purely technological limitations, not fundamental reasons like galaxies leaving the Hubble sphere (there's not really anything special about the distance where recession velocities reach $c$) or crossing the cosmic event horizon (galaxies get frozen in time while approaching the horizon instead of crossing it).
The distance to the farthest galaxy that will be observable of course changes with metric expansion of space. If you go by Pulsar's answer to a related question, the size of the asymptotically observable universe will be about $60\mathrm{Gly}$ as measured in co-moving distance:

$^1$ If there exist galaxies beyond the cosmic event horizon (which is plausible), the light from beyond the horizon is on its way towards us - it just takes a longer-than-infinite time to arrive. In the picture above, there's nothing special about the region outside the red cone.
A: I have argued that the Universe is rotating rather than expanding by connecting the Newtonian velocity and acceleration to the cosmic velocity and acceleration respectively using a simple algebra. The argument for rotating Universe instead than expanding Universe is the following : 
$$v = \sqrt{\frac{G M}{r}} = \frac{r}{r} \sqrt{\frac{G M}{r}} = r \sqrt{\frac{G M}{r^3}} = r H$$
$$a = \frac{G M}{r^2} = \frac{r}{r} \frac{G M}{r^2} = r \frac{G M}{r^3} = r H^2 = v H$$
$$H^2 = \frac{G M}{r^3} = (G \cdot \text{mass density}) = t^{-2}$$
where $v$ is the Newtonian velocity, $a$ is the Newtonian acceleration, $v = r H$ is the Hubble law or the cosmic velocity, $a = r H^2 = v H$ is the cosmic acceleration affecting the light and the Pioneer spacecraft, and so on. 
This means that the Universe is rotating rather than expanding and hence there is no place for the expansion within the laws of gravity. In this scenario the cosmic redshift is due to the time dilation of light rather than to the recession of galaxies from each other. The expansion that the Holy Quran is talking about is the one that occured during the formation of the Universe and which has stoped early in cosmic history leading the place for the gravity. 
Second thing is that the speed of light is found to signify the escape velocity of light at the position of the Earth in the cosmic gravitational field from equalizing the light energy to the cosmic gravitational potential energy. The argument for the equivalence between the speed of light and the escape velocity of light in the cosmic gravitational field is as following : 
$$E = m  c^2 = \frac{G M m}{r}$$
which gives $$c = \sqrt{\frac{G M}{r}}$$
Consequently, we can use the equation $c = r H$ to derive the distance from the center of the Universe to the position of the Earth in the Universe, which is calculated to be around $1.28 \cdot 10^{26}$ meters or $13.80$ Billion light-years. This distance has nothing to do with the radius of the measured Universe nor the observable Universe as generally believed. 
The third point is that the thickness distance of the Universe is obtained by looking back in time to the instant of Big Bang to be around $63$ Billion light-years using the technique of comoving distance in theoretical cosmology. 
Thus, the distance separating the Earth to the end of the Universe could be calculated by subtracting the distance between the center of the Universe and the Earth which is equal to $1.28 \cdot 10^{26}$ meters or $13.80$ Billion light-years from the thickness distance of the Universe which is equal to $5.78 \cdot 10^{26}$ meters or $63$ Billion light-years. 
Following this approach, it is found that the distance separating the Earth to the boundary limit of the Universe is around $4.50 \cdot 10^{26}$ meters or $50$ Billion light-years, that is $r' = (5.78 - 1.28) \cdot 10^{26}$ meters =  $4.5 \cdot 10^{26}$ meters or $r' = (63 - 13.80)$ Billion light-years = $49.20$ Billion light-years. 
